Free Actions of Compact Quantum Groups on Unital $C^{\ast }$-Algebras

Abstract

Let $F$ be a field, $\Gamma$ a finite group, and $\mathrm{Map}(\Gamma ,F)$ the Hopf algebra of all set-theoretic maps $\Gamma \to F$. If $E$ is a finite field extension of $F$ and $\Gamma$ is its Galois group, the extension is Galois if and only if the canonical map $E{\otimes }_{F}E\to E{\otimes }_F\mathrm{Map}(\Gamma ,F)$ resulting from viewing $E$ as a $\mathrm{Map}(\Gamma ,F)$-comodule is an isomorphism. Similarly, a finite covering space is regular if and only if the analogous canonical map is an isomorphism. In this paper, we extend this point of view to actions of compact quantum groups on unital $C^{\ast }$-algebras. We prove that such an action is free if and only if the canonical map (obtained using the underlying Hopf algebra of the compact quantum group) is an isomorphism. In particular, we are able to express the freeness of a compact Hausdorff topological group action on a compact Hausdorff topological space in algebraic terms. As an application, we show that a field of free actions on unital $C^{\ast }$-algebras yields a global free action.